By Restore Mode are you trying to say "safe mode" in the windows world? You could always boot into init 3 which is the non X mode. Give some more details to what you're trying to accomplish
Init 3 wouldn't help much, probably just not start X, and that's not the problem here (if the screen blanks way before the boot processes end and X is tried to start). The correct way would be to enter single-user mode, where you get logged in as root right away, and I take it that it's the closest to a "rescue mode" you can get
Here's how it goes:
1) enter Grub menu
2) hilight the kernel you'd normally boot (default kernel)
3) press 'e' (instructions are at the bottom of the screen, btw) to edit the boot entry
4) hilight the line that starts with 'kernel'
5) alter the options in the line (words after the kernel word): add word 'single' (without quotes), and maybe replace 'splash' with 'nosplash'. The 'single' option causes it to boot into a single-user mode (runlevel 2 if I'm not mistaken), and 'nosplash' causes it not to use the regular graphical boot splash screen you'd normally see - instead it should show the boot texts which might help you detect problems.
From there on you should be in a text shell environment logged in as root, ready to fix problems. Note that you should set a password for Grub so that not everyone can just enter the single-user mode and gain root access immediately (and set a BIOS password to prevent them from booting a live-cd, and do a million other things to make it more secure).